ISDE with logarithmic interaction and characteristic polynomials

Abstract

We consider certain random matrix eigenvalue dynamics, akin to Dyson Brownian motion, introduced by Rider and Valko. We show that from every initial condition, including ones involving coinciding coordinates, the dynamics, enhanced with more information, converge on path-space to a new infinite-dimensional Feller-continuous diffusion process. We show that the limiting diffusion solves an infinite-dimensional system of stochastic differential equations (ISDE) with logarithmic interaction. Moreover, we show convergence in the long-time limit of the infinite-dimensional dynamics starting from any initial condition to the equilibrium measure, given by the inverse points of the Bessel determinantal point process. As far as we can tell, this is: (a) the first path-space convergence result of random matrix dynamics starting from every initial condition to an infinite-dimensional Feller diffusion, (b) the first construction of solutions to an ISDE with logarithmic interaction from every initial condition for which the singular drift term can be defined at time $0$, (c) the first convergence to equilibrium result from every initial condition for an ISDE of this kind. The argument splits into two parts. The first part builds on the method of intertwiners introduced and developed by Borodin and Olshanski. The main new ingredients are a uniform, in a certain sense, approximation theorem of the spectrum of a family of random matrices indexed by an infinite-dimensional space and an extension of the method of intertwiners to deal with convergence to equilibrium. The second part introduces a new approach towards convergence of the singular drift term in the dynamics and for showing non-intersection of the limiting paths via certain ``characteristic polynomials" associated to the process. We believe variations of it will be applicable to other infinite-dimensional dynamics coming from random matrices.